Shortly after Valeant Pharmaceuticals International Inc. bought Medicis Pharmaceutical Corp. two years ago, the new owner gathered staffers of the skin-products maker and started handing out envelopes. If you got a black envelope, it meant you were fired.

Most of the research-and-development staff at Medicis headquarters in Scottsdale, Ariz., got the black envelopes, according to people familiar with the matter. Soon, most of the...